Stability analysis of a prey refuge predator–prey model with Allee effects

Abstract

We constructed a discrete-time predator–prey model by adding prey refuge and Allee effects (predator saturation on prey and mate limitation on predator) to an earlier prey–predator model and examined its dynamics. We show the existence of positive fixed points and study the stability properties. The numerical simulations and bifurcation diagrams verify the impact of refuge and the Allee mechanism on the system.

Appendix

If a non-hyperbolic map is defined on ${\mathbb{R}}^{2}$, then its dynamics may be analyzed by studying the dynamics on an associated one-dimensional center manifold $M_{\text{c}}$ (Elaydi 2008). This theorem is applicable only for the fixed point (0,0). Hence, we get the following new system by setting $x_{t} = N_{t} - 1$ and $y_{t} = P_{t}$:

$$ \begin{array}{*{20}l} {x_{t + 1} = x_{t} - e^{{ - (\beta /1 + s(1 + x_{t} ))}} rx_{t} (1 + x_{t} ) - a(1 - d)(1 + x_{t} )y_{t} } \hfill \\ {y_{t + 1} = y_{t} + \frac{{a((1 - d)(1 + x_{t} ) - y_{t} )y_{t}^{2} }}{{m + y_{t} }}.} \hfill \\ \end{array} $$

(6)

Let $J^{*}$ be the Jacobian matrix of the system (6). Then

$$ J^{*} (0,0) = \left( {\begin{array}{*{20}c} {1 - re^{ - (\beta /1 + s)} } & { - a(1 - d)} \\ 0 & 1 \\ \end{array} } \right) $$

Rewriting system (6), we obtain

$$ \begin{array}{*{20}l} {x_{t + 1} = (1 - re^{ - (\beta /1 + s)} )x_{t} - ay_{t} (1 - d) + \tilde{f}(x_{t} ,y_{t} )} \hfill \\ {y_{t + 1} = y_{t} + \tilde{g}(x_{t} ,y_{t} ),} \hfill \\ \end{array} $$

(7)

where

$$ \tilde{f}(x,y) = x({\text{re}}^{ - (\beta /1 + s)} - {\text{re}}^{ - (\beta /1 + s + sx)} (1 + x) + a( - 1 + d)y), $$

and

$$ \tilde{g}(x,y) = \frac{{a((1 - d)(1 + x) - y)y^{2} }}{m + y}. $$

By using the Taylor series expansion of $e^{ - (\beta /1 + s + sx)}$ at the point $x = 0$, we approximate

$$ e^{ - (\beta /1 + s + sx)} \approx e^{ - (\beta /1 + s)} + \frac{{ae^{ - (\beta /1 + s)} sx}}{{(1 + s)^{2} }} + \frac{{\beta e^{ - (\beta /1 + s)} ( - 2 + \beta - 2s)s^{2} x^{2} }}{{2(1 + s)^{4} }} + O(x^{3} ) $$

For simplification, let $k = e^{ - (\beta /1 + s)}$. Then

$$ \tilde{f}(x,y) \approx x\left( {kr - \left( {k + \frac{\beta ksx}{{(1 + s)^{2} }} + \frac{{\beta k( - 2 + \beta - 2s)s^{2} x^{2} }}{{2(1 + s)^{4} }}} \right)r(1 + x) + a( - 1 + d)y} \right). $$

Since the invariant manifold is tangent to the corresponding eigenspace by the Invariant Manifold Theorem, assume that the map $h$ takes the form

$$ h(y) = \frac{a(d - 1)}{rk}y + c_{1} y^{2} + c_{2} y^{3} + O(y^{4} ). $$

To compute $c_{1}$ and $c_{2}$, the following functional equation should be solved:

$$ h(y + \tilde{g}(h(y),y)) \approx (1 - re^{ - (\beta /1 + s)} )h(y) - a(1 - d)y + \tilde{f}(h(y),y). $$

We have

$$ c_{1} \approx - \frac{{a^{2} ( - 1 + d)^{2} \left( { - 1 - 2s - s^{2} + ms\beta } \right)}}{{k^{2} mr^{2} (1 + s)^{2} }}, $$

and

$$ \begin{aligned} c_{2} \approx & \frac{{a^{2} ( - 1 + d)}}{{2k^{3} m^{2} r^{3} (1 + s)^{4} }}(2k(1 - d + m)r(1 + s)^{4} \\ & + a( - 1 + d)^{2} (4(1 + s)^{4} + 2ms(1 + s) \\ & ( - 4 + ( - 4 + m)s)\beta + 3m^{2} s^{2} \beta^{2} )). \\ \end{aligned} $$

This leads to the equation

$$ \begin{aligned} P(y) \approx & y - \frac{{a( - 1 + d)y^{2} }}{m} - \frac{{a\left( {a( - 1 + d)^{2} m + k(1 - d + m)r} \right)y^{3} }}{{km^{2} r}} \\ & + \frac{{a\left( {1 - d + m + (a( - 1 + d)^{2} m/kr) - (a^{2} ( - 1 + d)^{3} m\left( {(1 + s)^{2} - ms\beta } \right)/k^{2} r^{2} (1 + s)^{2} )} \right)y^{4} }}{{m^{3} }}. \\ \end{aligned} $$

We have $P^{\prime}(0) = 1$ and $P^{\prime \prime } (0) = 2a(1 - d)/m > 0$ for $a > 0$, $m > 0$ and $d \in [0,1)$. Hence, by Theorem 1.5 in Elaydi (2008), (1, 0) is a semistable fixed point if $\beta /(1 + s) > \ln (r/2)$ (See figure 13).

Figure 13

The map P on the semistable invariant manifold $x = h(y)$ for $\beta = 4$, $a = 2$, $r = 5$, $s = 1$, $m = 2$ and $d = 0.4$.